Splitting a stationary set concentrated on regular cardinals

Let $T=S\setminus \mathrm{Tr}(S)$, stationary. For each $\alpha \in T$ use AC to choose a club $D_{\alpha }\subseteq \alpha$ disjoint from $S\cap \alpha$, and let $c^{\alpha }:\alpha \to D_{\alpha }$ be its normal enumeration. Such clubs exist since alpha is regular uncountable and is not in the trace.

For each coordinate $\xi <\kappa$ consider the domain $T_{\xi }=\{\alpha \in T:\xi <\alpha \}$. Suppose no coordinate has stationary sets $\{\alpha \in T_{\xi }:c^{\alpha }(\xi )\ge b\}$ for every $b<\kappa$. Choose a failing threshold $b_{\xi }$ and avoiding club $C_{\xi }$, so $c^{\alpha }(\xi )<b_{\xi }$ whenever $\alpha \in T_{\xi }\cap C_{\xi }$.

Let $D={<mi\; mathvariant="normal">△</mi>}_{\xi <\kappa }{C}_{\xi }$ and let $E$ be the club of closure points of $\xi \mapsto b_{\xi }$. Choose $\alpha \in T\cap E$ and then $\gamma \in T\cap D$ above alpha. For every $\xi <\alpha$, diagonal membership of gamma and closure at alpha give $c^{\gamma }(\xi )<b_{\xi }<\alpha$. Since alpha is a nonzero limit, continuity gives $c^{\gamma }(\alpha )\le \alpha$. Strict increase implies $c^{\gamma }(\xi )\ge \xi$ by ordinal induction, so in fact $c^{\gamma }(\alpha )=\alpha$. This contradicts $D_{\gamma }\cap S=\mathrm{\varnothing }$, since $\alpha \in T\subseteq S$.

Consequently some $\xi$ has all stationary tail domains. Its coordinate map on $T_{\xi }$ is regressive and the fibre lemma partitions $T_{\xi }$ into kappa stationary sets. Add $S\setminus T_{\xi }$ to one piece; this preserves stationarity and disjointness and gives the desired partition of S.